Maybe you start by showing me how it is a philosophical problem? I have no idea what the problem is you are seeing.
Slow down. First, i do see it is a theoretical problem, and I greatly appreciate the level of factual intelligent response that has been given. For the most part (perhaps minus my own comments?) we have been painting the theoretical landscape (and its been great).
The problem in question isn't lines. Its the problem of "straight" lines. Straight is descriptive, notional, theoretical, conceptual, where "line" is definite, non abstract, material, perceptible, etc.
If I asked you to apply the notion of straight, say, to your belief in science, you would wisely answer "it is as straight as the verity of its claims" and not "science is straightness", since by this second statement, you would be reversing the nature of science itself. Ex. We don't need kings, we need good kings.
Thus the notion of straight must also be applied conceptually to "line", where the "line" itself is neither straight nor curved. It is merely a succession of locations, implying an undefined direction and form.
I think the implications are philosophical in nature, since the term "straight" is conceptual in nature, and philosophy is largely the science of Being, which involves the way we conceive, perceive, even deceive our knowledge.
I think we're all intrigued by opinion on the question, but if anyone feels that the mathematical theoretics are still in need of clarification, my original interest was in further wrapping my own head around the inherent nature of the problem.
Fun right?